Calculating the Height of a Pole Using Trigonometry and Angles

Introduction

Trigonometry is a powerful tool in solving real-world problems, including those related to heights and distances. In this article, we will demonstrate how to calculate the height of a pole using trigonometric principles, specifically angles of elevation and depression. We will use a practical example as a case study and employ step-by-step calculations to find the solution.

Case Study: Height of a Pole

Consider a scenario where a person of height 5 feet observes the angle of depression to the base of a pole as 30 degrees and the angle of elevation to the top of the pole as 60 degrees. The goal is to find the height of the pole.

Step 1: Setting up the Situation

Let's denote:

H:the height of the pole D: the horizontal distance from the person to the base of the pole Height of the person: 5 ft.

The angle of depression to the base of the pole is 30 degrees, meaning the angle down from the person's eye level to the base of the pole is 30 degrees. Conversely, the angle of elevation to the top of the pole is 60 degrees.

Step 2: Finding the Horizontal Distance (D)

We can use the tangent function for the angle of depression:

tan(30°) frac{text{opposite}}{text{adjacent}} frac{5}{D}

Since tan(30°) frac{1}{sqrt{3}} , we have:

frac{1}{sqrt{3}} frac{5}{D}

Rearranging this equation gives:

D 5sqrt{3}

Step 3: Finding the Height of the Pole (H)

Using the angle of elevation to the top of the pole:

tan(60°) frac{text{opposite}}{text{adjacent}} frac{H - 5}{D}

Since tan(60°) sqrt{3} , we have:

sqrt{3} frac{H - 5}{5sqrt{3}}

Rearranging this equation gives:

H - 5 5sqrt{3} cdot sqrt{3} 15

Therefore:

H 15 5 20 text{ ft}

Conclusion

The height of the pole is 20 feet.

Alternatively, if the sight line to the pole is 12 meters and the line to the top of the pole forms a 30-60-90 triangle, the distance from the sight line to the top of the pole is 12sqrt{3} meters. Calculating the horizontal distance and the additional height gives the total height as approximately 27.7 meters. Breaking it down:

Height of pole h1 h2 20.78m 6.93m 27.71m h1 tan(60°) cdot 12m 20.78m h2 tan(30°) cdot 12m 6.93m

Conclusion

Using trigonometry effectively, along with the principles of angles of elevation and depression, we can solve real-world problems, such as determining the height of a pole. This method provides a clear and precise solution to such geometric challenges.